The regular OT problem defined in (6) can be solved by an effective linear programming method in the order of \(\mathcal {O}(n^3 log (n))\)  time complexity, where \(n\)  is number of the points in each probability measures. Cuturi [1]} has introduced an interesting approach which relaxes the OT problem by adding a strong convex regularizer to the OT cost function to reduce the time complexity to \(\mathcal {O}(n^2)\) . Specifically, this approach asks for a solution \(T^{\prime }\)  with more entropy, instead of computing the exact Wasserstein distance. In other words, the regularized OT
distances can interpolate the solution, depending
on the regularization strength \(\gamma \) , between exact OT (\(\gamma =0\)  ), and Maximum Mean Discrepancy, MMD, (\(\gamma =\infty \) ). In this work, we use the regularized OT not only for the matter of time complexity, but also it has been shown that the sample complexity of exact Wasserstein distance is \(O(1/n^{1/d})\) , while the regularized Wasserstein distance depending on \(\gamma \)  value, is between \(O(1/\sqrt{n})\)  and \(O(1/n^{1/d})\) , where \(d\)  is dimension of the samples [2]}, [3]}. This means that the entropic regularization reduces the chance of over-fitting for our SSL model when it computes the Wasserstein distance between output of the CNN obtained from the labeled and unlabeled data. Hence, our OT problem in the regularized form is recast as follows:
\(\tilde{W}_{\gamma }(\mathcal {M^{\prime }}, \mathcal {M})=\underset{T^{\prime } \in P^{\prime }(a, \alpha )}{\mathrm {min}} \big \langle T^{\prime }, X\big \rangle -\gamma E(T^{\prime }),\) 
